Department of Mathematics and

Department of Economics

University of British Columbia

**Abstract**: We will be investigating some basic questions
economic theory: do individuals behave as the theory predicts, that is, do they
maximize some utility function? If so, can their preferences be inferred from
observed behaviour? We will show this is the case for individuals, and then we
will turn to couples and finally to markets. Answering these questions requires
mathematical tools which were developed by Elie Cartan and others for the
purpose of studying systems of partial differential equations arising in
geometry, but which now turn out to be crucial in other settings. We will spend
about half the time explaining these methods, and half the time applying them
to economic situations

**Lecture 1**:- Motivation: characterization, identification, the
unitary model, the Slutsky relations
**Lecture 2**:- Differential forms, exterior products and derivatives,
theorems of Frobenius, Darboux, and Ekeland-Nirenberg
**Lecture 3**:- Households, the efficiency assumption, the
Browning-Chiappori condition, characterization and identification
**Lecture 4**:- Exterior differential systems, integral manifolds, the
Cartan-Kähler theorem
**Lecture 5**:- Markets with many agents: excess demand (Sonnenschein-Mantel-Debreu theorem), market demand (Chiappori-Ekeland theorem)